48,718 research outputs found
P-splines with derivative based penalties and tensor product smoothing of unevenly distributed data
The P-splines of Eilers and Marx (1996) combine a B-spline basis with a
discrete quadratic penalty on the basis coefficients, to produce a reduced rank
spline like smoother. P-splines have three properties that make them very
popular as reduced rank smoothers: i) the basis and the penalty are sparse,
enabling efficient computation, especially for Bayesian stochastic simulation;
ii) it is possible to flexibly `mix-and-match' the order of B-spline basis and
penalty, rather than the order of penalty controlling the order of the basis as
in spline smoothing; iii) it is very easy to set up the B-spline basis
functions and penalties. The discrete penalties are somewhat less interpretable
in terms of function shape than the traditional derivative based spline
penalties, but tend towards penalties proportional to traditional spline
penalties in the limit of large basis size. However part of the point of
P-splines is not to use a large basis size. In addition the spline basis
functions arise from solving functional optimization problems involving
derivative based penalties, so moving to discrete penalties for smoothing may
not always be desirable. The purpose of this note is to point out that the
three properties of basis-penalty sparsity, mix-and-match penalization and ease
of setup are readily obtainable with B-splines subject to derivative based
penalization. The penalty setup typically requires a few lines of code, rather
than the two lines typically required for P-splines, but this one off
disadvantage seems to be the only one associated with using derivative based
penalties. As an example application, it is shown how basis-penalty sparsity
enables efficient computation with tensor product smoothers of scattered data
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
Fused kernel-spline smoothing for repeatedly measured outcomes in a generalized partially linear model with functional single index
We propose a generalized partially linear functional single index risk score
model for repeatedly measured outcomes where the index itself is a function of
time. We fuse the nonparametric kernel method and regression spline method, and
modify the generalized estimating equation to facilitate estimation and
inference. We use local smoothing kernel to estimate the unspecified
coefficient functions of time, and use B-splines to estimate the unspecified
function of the single index component. The covariance structure is taken into
account via a working model, which provides valid estimation and inference
procedure whether or not it captures the true covariance. The estimation method
is applicable to both continuous and discrete outcomes. We derive large sample
properties of the estimation procedure and show a different convergence rate
for each component of the model. The asymptotic properties when the kernel and
regression spline methods are combined in a nested fashion has not been studied
prior to this work, even in the independent data case.Comment: Published at http://dx.doi.org/10.1214/15-AOS1330 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Error analysis for quadratic spline quasi-interpolants on non-uniform criss-cross triangulations of bounded rectangular domains
Given a non-uniform criss-cross partition of a rectangular domain ,
we analyse the error between a function defined on and two types
of -quadratic spline quasi-interpolants (QIs) obtained as linear
combinations of B-splines with discrete functionals as coefficients. The main
novelties are the facts that supports of B-splines are contained in
and that data sites also lie inside or on the boundary of . Moreover,
the infinity norms of these QIs are small and do not depend on the
triangulation: as the two QIs are exact on quadratic polynomials, they give the
optimal approximation order for smooth functions. Our analysis is done for
and its partial derivatives of the first and second orders and a particular
effort has been made in order to give the best possible error bounds in terms
of the smoothness of and of the mesh ratios of the triangulation
Cardinal Exponential Splines: Part II—Think Analog, Act Digital
By interpreting the Green-function reproduction property of exponential splines in signal processing terms, we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts. The link is that the latter are the B-spline coefficients of the former (which happen to be exponential splines). Motivated by this observation, we introduce an extended family of cardinal splines—the generalized E-splines—to generalize the concept for all convolution operators with rational transfer functions. We construct the corresponding compactly-supported B-spline basis functions, which are characterized by their poles and zeros, thereby establishing an interesting connection with analog filter design techniques. We investigate the properties of these new B-splines and present the corresponding signal processing calculus, which allows us to perform continuous-time operations, such as convolution, differential operators, and modulation, by simple application of the discrete version of these operators in the B-spline domain. In particular, we show how the formalism can be used to obtain exact, discrete implementations of analog filters. Finally, we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real-world analog-to-digital (A-to-D) and D-to-A conversion systems
Isotropic Polyharmonic B-Splines: Scaling Functions and Wavelets
In this paper, we use polyharmonic B-splines to build multidimensional wavelet bases. These functions are nonseparable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic B-splines do not converge to a Gaussian as the order parameter increases, as opposed to their separable B-spline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic B-splines. Next, we focus on the two-dimensional quincunx subsampling scheme. This configuration is of particular interest for image processing, because it yields a finer scale progression than the standard dyadic approach. However, up until now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from the scaling functions, which are the polyharmonic B-splines and, as such, explicitly known, and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi-orthogonal wavelet transform; e.g., orthonormal, B-spline, and dual. The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic B-spline wavelet converges to a combination of four Gabor atoms, which are well separated in the frequency domain. We also show that these wavelets are nearly isotropic and that they behave as an iterated Laplacian operator at low frequencies. We describe an efficient fast Fourier transform-based implementation of the discrete wavelet transform based on polyharmonic B-splines
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